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# P2.T5.113. Monte Carlo for valuation of mortgage-backed securities (MBS)

#### David Harper CFA FRM

##### David Harper CFA FRM
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AIMs: Discuss the steps for valuing a mortgage security using Monte Carlo methodology. Explain how to select the number of interest rate paths in Monte Carlo analysis. Define reinvestment risk.

Questions:

113.1. We can use the following four generic steps to value a mortgage-backed security (MBS) with Monte Carlo simulation. Step 1: Simulate short-term interest rate and refinancing rate paths; Step 2: Project the cash flow on each interest rate path; Step 3: Determine the present value of the cash flows on each interest rate path; Step 4: Compute the theoretical value of the mortgage security. Each of the following is TRUE about the process, EXCEPT for the false statement which is?

a. In Step 1, many vendors use both short and long yield volatilities (or even a term structure), where the short yield volatility is assumed to be greater than the long yield volatility
b. In Step 2, the most common approach is to assume a constant, predetermined prepayment rate across the paths; e.g., 150% PSA for all paths (T) and months (N).
c. In Step 3, the discount rate for determining the present value is the simulated spot rate for each month on the interest rate path plus an appropriate spread; the spot rate on a path can be determined from the simulated future monthly rates.
d. In Step 4, the present value of a given interest rate path is the theoretical value of a pass-through if that path was actually realized; and the theoretical value of the pass-through is the average of the theoretical values of all the interest rate paths.

113.2. Your colleague Jane is using Monte Carlo simulation to value a mortgage-backed security. She asks for your advice on selecting the number of interest rate paths. Each of the following is valid (true) advice EXCEPT for:

a. She can increase the number of paths but the simulation's accuracy will only increase roughly with the square root of the number of paths; i.e., to double accuracy requires about 4x paths
b. She can use techniques to reduce the variance (variance reduction)
c. Instead of a full Monte Caro simulation, she can employ representative paths. Representative paths should be more accurate but at the cost of additional trials (paths)
d. If she uses the representative paths technique, the theoretical value of the security is then the weighted average of the representative paths; the weight for a path is the percentage of that representative path relative to the total sample paths.

113.3. Roger participates in your FRM study group and he just read the Fabozzi assignment called "Techniques for Valuing Mortgage-Backed Securities (MBS)." Based on his reading, he makes the following assertions about reinvestment risk:

I. The cash flow yield (a.k.a., mortgage yield) is designed to overcome the failure of the bond-equivalent yield to incorporate reinvestment risk
II. A mortgage-backed security (MBS) has less reinvestment risk, than typical fixed-income investments, due to prepayments
III. If we hold the MBS to maturity, then we can avoid reinvestment risk and ensure that realized return equals the original yield
IV. If all other things are equal (ceteris paribus), a higher MBS pass-through coupon rate implies less (lower) prepayment risk

Which of Roger's statements is (are) correct?

a. None are correct, Roger needs to re-read the chapter
b. Only I. is correct, Roger could use a lot more studying
c. Only III. and IV. are correct, Roger could use a little more studying
d. All are correct, Roger sounds like he's read the assignment

Answers: